1.7.2 The Event-Study Methodology

Basic Prerequisites: Econometrics & Time Series (OLS regression, stationarity) and 01 · From Zero.

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1. Intuition & Practical Objective

The event-study design is a precise timeline around a corporate announcement. The practical objective of this page: know exactly which windows exist, why each has its length, and how the market model converts raw returns into abnormal returns by estimating a firm’s sensitivity to the market out of sample.

Three windows (Brown–Warner 1985; Kothari–Warner 2007):

1. Estimation window (e.g. days $-244$ through $-6$, ~239 observations): a quiet pre-event stretch used to estimate the normal-return model’s parameters. It must end before the event so the parameters are uncontaminated.
2. Event window (e.g. days $-5$ through $+5$): the days whose abnormal returns are measured. It brackets the announcement to catch pre-event leakage (partial anticipation) and post-event adjustment.
3. Post-event window (optional, days $+1$ onward): used to test efficiency - does the abnormal return persist after the announcement? (Persistent drift is inconsistent with efficiency.)

The market model (Sharpe’s single-index regression) is the standard expected-return model:

$$R_{it}={\alpha }_i+{\beta }_i{R}_{mt}+{\epsilon }_{it},$$

estimated by OLS in the estimation window. The abnormal return on each event-window day is then the prediction error:

$$A{R}_{it}=R_{it}-({\hat{\alpha }}_i+{\hat{\beta }}_i{R}_{mt}).$$

Because daily expected returns are ~0.05% (Kothari–Warner §4.2), even a substantially misestimated $\beta$ barely changes a short-window abnormal return - which is precisely why short-horizon event studies are robust to the expected-return model (Brown–Warner 1985).

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2. Mathematical Ground Truth & Derivations

The market model. Under the market model, the firm’s return is a linear function of the market return plus an idiosyncratic shock ${\epsilon }_{it}$:

$$R_{it}={\alpha }_i+{\beta }_i{R}_{mt}+{\epsilon }_{it},\mathbb{E}[{\epsilon }_{it}]=0,\mathrm{Cov}({\epsilon }_{it},R_{mt})=0.$$

OLS in the estimation window gives the prediction for the event window:

$${\hat{\beta }}_i=\frac{\mathrm{Cov}(R_i,R_m)}{\mathrm{Var}(R_m)},{\hat{\alpha }}_i={\bar{R}}_i-{\hat{\beta }}_i{\bar{R}}_m.$$

Abnormal return = prediction error (Kothari–Warner eq. 1–2; Brown–Warner eq. 4):

$$A{R}_{it}=R_{it}-({\hat{\alpha }}_i+{\hat{\beta }}_i{R}_{mt}),t\in \text{event window}.$$

Alternative expected-return models (Brown–Warner 1985, §3.2):

• Mean-adjusted: $K_{it}={\bar{R}}_i$ (the firm’s own estimation-window average) - simplest, robust.
• Market-adjusted: $K_{it}=R_{mt}$ (expected return = the market return; forces $\alpha =0,\beta =1$) - useful when the estimation window is too short for OLS.

Cross-sectional aggregation (Kothari–Warner eq. 3). The mean abnormal return at event time $t$ across $N$ sample firms:

$$A{R}_t=\frac{1}{N}\sum _{i=1}^{N}A{R}_{it}.$$

The event window’s length is a design trade-off: a longer window (say $-10,+10$) captures slow price adjustment and partial anticipation, but admits more noise and contamination; a tight window (say $0,+1$) maximizes power when the announcement date is precise (e.g. earnings announcements - Kothari–Warner §3.6).

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3. Computational Implementation - the full market-model pipeline

Stdlib only. Estimates the market model in the estimation window, computes the event-day abnormal return, the CAR, and the t-statistic. The day-0 noise is zeroed so the injected +3% event is the only abnormal signal - a single firm, so the event-day t is modest (idiosyncratic noise dominates one firm; aggregation across firms fixes that, see 03).

Reading the output. The OLS market model recovers $\hat{\beta }=1.16$ close to the true $1.2$; the event-day abnormal return is $+3.02\%$ (the injected signal), but its t-statistic is only $+1.46$ - one firm’s signal is swamped by its own noise (the neighbor days $-1,+1$ happen to be negative, so the 3-day CAR is negative). This is exactly why real event studies average over many firms (03), which restores the statistical signal.

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4. Failure Modes & First-Principles Breakdowns

1. Estimation window contaminated by the event. If the estimation window overlaps the announcement (or contains other value-relevant news), $\hat{\alpha },\hat{\beta }$ are biased and every abnormal return inherits the bias. Keep the estimation window clean and ending before the event.
2. Event-window contamination. A second value-relevant event inside the event window (earnings + merger same week) gets attributed to the event of interest. This is the single most common source of biased CARs (05).
3. Non-synchronous trading. With daily data, a thinly-traded stock’s return and the market return are measured over different intervals, so OLS $\hat{\beta }$ is biased and inconsistent (Brown–Warner 1985 §2.2; Scholes–Williams 1977). Mitigate with lag/lead corrections.
4. Wrong window length. Too long → noise and contamination dilute the signal (low power); too tight → you miss slow adjustment or pre-event leakage (partial anticipation shows up before day 0). Match the window to how precisely the announcement date is known.

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5. References

• Brown & Warner (1985), Using Daily Stock Returns: The Case of Event Studies, JFE 14(1)
• Kothari & Warner (2007), Econometrics of Event Studies, Handbook of Corporate Finance
• MacKinlay (1997), Event Studies in Economics and Finance, JEL 35(1)
• Sharpe (1964), Capital Asset Prices, J. Finance 19(3)

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Abnormal Returns & CAR · 04 · Statistical Testing
• Base: Econometrics & Time Series (OLS, the market-model regression)